Tài liệu 218023_dynamic system and control lecture 3_updated

Dynamic Systems and Control, Chapter 3: Feedback Control Theory Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-1 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Lịch học bù: Ngày: 21/3/2015 Phòng: 211 B1 Thời gian: Tiết 1-2 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-2 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Root-Locus Method - - The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. If the system has a variable loop gain, then the location of the closedloop poles depends on the value of the loop gain chosen. The closed-loop poles are the roots of the characteristic equation. Finding the roots of the characteristic equation of degree higher than 3 is laborious and will need computer solution (Matlab can do it). However, just finding the roots of the characteristic equation may be of limited value, because as the gain of the open-loop transfer function varies, the characteristic equation changes and the computations must be repeated. Root-locus method, is one in which the roots of the characteristic equation are plotted for all values of a system parameter By using the root-locus method the designer can predict the effects on the location of the closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zeros © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-3 Dynamic Systems and Control, Chapter 3: Feedback Control Theory ROOT-LOCUS PLOTS Angle and Magnitude Conditions Consider the negative feedback system The characteristic equation for this closed-loop system Angle condition Magnitude condition © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-4 Dynamic Systems and Control, Chapter 3: Feedback Control Theory ROOT-LOCUS PLOTS When G(s)H(s) involves a gain parameter K, characteristic equation may be written as Then, the root loci for the system are the loci of the closedloop poles as the gain K is varied from zero to infinity © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-5 Dynamic Systems and Control, Chapter 3: Feedback Control Theory RELATIONSHIP BETWEEN ZEROS-POLES AND ANGLEMAGNIGTUDE The angle of G(s)H(s) is The magnitude of G(s)H(s) for this system is © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-6 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Consider the control system Illustrative example The characteristic equation Rearrange this equation in the form © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-7 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 1. Locate the poles and zeros of G(s)H(s) on the s plane. The root-locus branches start from open-loop poles and terminate at zeros (finite zeros or zeros at infinity). - The root loci are symmetrical about the real axis of the s plane, because the complex poles and complex zeros occur only in conjugate pairs. - If the number of closed-loop poles is the same as the number of open-loop poles, then the number of individual root-locus branches terminating at finite open-loop zeros is equal to the number m of the open-loop zeros. The remaining n-m branches terminate at infinity (n-m implicit zeros at infinity) along asymptotes Illustrative example The first step in constructing a rootlocus plot is to locate the open-loop poles, s = 0, s =–1, and s =–2, in the complex plane © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-8 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 2. Determine the root loci on the real axis. Root loci on the real axis are determined by open-loop poles and zeros lying on it. - Each portion of the root locus on the real axis extends over a range from a pole or zero to another pole or zero. - In constructing the root loci on the real axis, choose a test point on it. If the total number of real poles and real zeros to the right of this test point is odd, then this point lies on a root locus. Illustrative example Q: - If the test point is on the positive real axis, then - If a test point on the negative real axis between 0 and –1, then - If a test point is selected between –1 and –2, then © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-9 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 3. Determine the asymptotes of root loci. The root loci for very large values of s must be asymptotic to straight lines whose angles (slopes) are given by Illustrative example Since the angle repeats itself as k is varied, the distinct angles for the asymptotes are determined as 60°, –60°, and 180° © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-10 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 3 (continued). Find the point where they intersect the real axis. The abscissa of the intersection of the asymptotes and the real axis is then obtained by Illustrative example The three straight lines shown are the asymptotes. They meet at point s = –1 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-11 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 4. Find the breakaway and break-in points. Because of the conjugate symmetry of the root loci, the breakaway points and break-in points either lie on the real axis or occur in complex-conjugate pairs. - If a root locus lies between two adjacent open-loop poles on the real axis, then there exists at least one breakaway point between the two poles. - If the root locus lies between two adjacent zeros (one zero may be located at – q) on the real axis, then there always exists at least one break-in point between the two zeros. - If the root locus lies between an open-loop pole and a zero (finite or infinite) on the real axis, then there may exist no breakaway or break-in points or there may exist both breakaway and break-in points. - Breakaway and break-in points can be determined from the roots of - It should be noted that not all the solutions of dK/ds = 0 correspond to actual breakaway points. If a point at which dK/ds = 0 is on a root locus, it is an actual break away or break-in point. Stated differently, if at a point at which dK/ds = 0 the value of K takes a real positive value, then that point is an actual breakaway or break-in point © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-12 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 4. (continued) Illustrative example © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-13 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 5. Determine the angle of departure (angle of arrival) of the root locus from a complex pole (at a complex zero). To sketch the root loci with reasonable accuracy, we must find the directions of the root loci near the complex poles and zeros. If a test point is chosen and moved in the very vicinity of a complex pole (or complex zero), the sum of the angular contributions from all other poles and zeros can be considered to remain the same. - Angle of departure from a complex pole= 180° – (sum of the angles of vectors to a complex pole in question from other poles) ± (sum of the angles of vectors to a complex pole in question from zeros) - Angle of arrival at a complex zero= 180° – (sum of the angles of vectors to a complex zero in question from other zeros) ± (sum of the angles of vectors to a complex zero in question from poles) There are no complex pole/zero in the present example © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-14 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 6. Find the points where the root loci may cross the imaginary axis. The points where the root loci intersect the j axis can be found easily by (i) use of Routh’s stability criterion or (ii) letting s = j in the characteristic equation, equating both the real part and the imaginary part to zero, and solving for  and K. The values of  thus found give the frequencies at which root loci cross the imaginary axis. The K value corresponding to each crossing frequency gives the gain at the crossing point. Illustrative example For the present example © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-15 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule 7. Determine closed-loop poles. A particular point on each root-locus branch will be a closed-loop pole if the value of K at that point satisfies the magnitude condition. Conversely, the magnitude condition enables us to determine the value of the gain K at any specific root location on the locus. (If necessary, the root loci may be graduated in terms of K. The root loci are continuous with K.) - The value of K corresponding to any point s on a root locus can be obtained using © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-16 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Draw the root loci © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-17 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Comments on the Root-Locus Plots - A slight change in the pole–zero configuration may cause significant changes in the root-locus configurations © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-18 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Cancellation of Poles of G(s) with Zeros of H(s) It is important to note that if the denominator of G(s) and the numerator of H(s) involve common factors, then the corresponding open-loop poles and zeros will cancel each other, reducing the degree of the characteristic equation by one or more. Example: The closed-loop transfer function The characteristic equation is Because of the cancellation of the terms (s+1) To obtain the complete set of closed-loop poles, we must add the canceled pole of G(s)H(s) to those closed-loop poles The reduced characteristic equation is obtained from the root-locus plot of G(s)H(s) © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-19 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Constant  loci and constant n loci The damping ratio  of a pair of complex-conjugate poles can be expressed in terms of the angle , which is measured from the negative real axis © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 3-20